Stand aside, Alberti and Brunelleschi, Palladio and Wren, Richardson and Sullivan, Wright and Mies. Your time has come and gone. If we are to read a message into this book, we are now in a new era of architecture, one in which mathematics is to play a much greater role than in the past.

To be sure, mathematics has always been a part of architecture, in measurements, calculating the strength of materials, and assessing stress. If we are to believe accounts of the construction of a building like the Parthenon, the mathematics was rather sophisticated — varying subtly the size and location of columns, for example, to give the impression of their being the same visually when in fact they are not. Later in the Renaissance, perspective and symmetry played a major role and were the basis of many discussions of the aesthetic elements of architecture. Of course, there has always been an engineering side to the field; no one wants buildings to fall down. With the introduction of steel structural materials, we got skyscrapers. And now with computer graphics there are many possibilities for daring, even what may seem to be reckless, designs; for example, bizarrely placed units stuck together in unexpected ways (Daniel Libeskind), huge wavy sheets of metal surrounding what are often essentially boxes (Frank Gehry), and gravity defying jokes like columns that instead of being set on a base to hold up the roof, instead hang from the ceiling and never touch the ground (Peter Eisenman).

In many cases these new structures might be mistaken for something mathematical, but in fact they are mostly virtuoso displays of engineering prowess. The stunningly beautiful bridges and train stations of Santiago Calatrava are good examples of masterful engineering. But the mathematics involved is engineering mathematics, not necessarily the kind that would be of much interest to most mathematicians. One problem with these new structures is that they usually cannot be built using parts off the shelf. They require specially designed structural elements, causing the costs to run very high and in many cases showing weaknesses within a few years. Joints don’t fit properly so buildings leak or quickly corrode. In the case of Gehry’s Disney Concert Hall, the reflections off the metal panels initially blinded passing motorists. And to accommodate the aesthetics of the exterior, the interiors suffer — in the case of the Disney, from some oppressively low ceilings. There’s also the occasional disorientation resulting, in the case of Gehry and Eisenman, in confusion over how to find the front door.

The buildings featured in this book are quite different (though Gehry’s Disney Hall does make a brief appearance, but the reason for its inclusion is not convincing — mention of Gaussian curvature and developable surfaces does not explain much about this building). Unfortunately, many will be subject to the flaws of the previous generation of buildings. But the new structures shown in this volume are at least related to real mathematics — topology, real analysis, minimal surfaces, graph theory, hyperbolic geometry, tilings, knot theory, among others. Of course, they require powerful engineering skills to bring them off the drawing board (or, more likely, off the computer screen). They involve Möbius strips, the Klein bottle, minimal surfaces (catenoids, helicoids, and the Enneper surface), sophisticated tessellations on the inner and outer surfaces (Sierpiński’s sieves, for example), Voronoi diagrams, and many more.

The mathematical descriptions are not for the faint of heart. It helps to have a reasonable mathematical background just to figure out what the mathematics is about, as well as to know how it is used. The authors are both architects at the RMIT University (Royal Melbourne Institute of Technology), but they appear to have a good grasp of where the mathematics comes into the design of the buildings they choose to discuss. At the same time, there are some descriptions that would not sound quite the way they do had they been written by professionals in mathematics. This may be intentional, however, since the audience for the book will clearly not be limited to mathematicians.

Because the authors are Australian, their choice of buildings may be somewhat skewed towards Australian projects, or to buildings in Asia, or in the cluster of oil-rich countries in the Middle East. The last is not surprising since that’s where architects can find clients who can afford these most extraordinarily expensive structures. The authors, to their credit, avoid pursuing the vulgar quest for yet another skyscraper a few stories higher than the last. The structures described in detail in the book are broken down geographically as follows: continental Europe (12), Australia (11), the United Kingdom (8), Abu Dhabi/Qatar (5), the Far East (4), and the United States (4). The United States may have some catching up to do.

After a few opening definitions of words that will be used in the text the authors launch right into a chapter on Minifie Nixon’s Australian Wildlife Health Centre near Melbourne. It’s clearly in the form of a Costa-Hoffman-Meeks (CHM) minimal surface. And a gorgeous building it is: a golden surface, of genus 3, of course, where the holes provide oculi for bringing natural light into the courtyard and acting as solar chimneys. “It opens on the opposite side of the building fabric to create three skylights that are evenly distributed around the main ambulatory viewing space.” The façade of the building is of masonry and just as a lagniappe, the architect has used it to form a cellular automaton.

In order to tie the exposition of the various buildings to the past, the authors discuss what they appear to view as the progenitor of such architecture, Antoni Gaudí’s Sagrada Familia in Barcelona. Gaudí had to make do with the repertoire of materials and techniques that he had available to him. And he did it with great effect. He did use ruled surfaces. But the CHM minimal surface was not known, for example, until 1984, long after Gaudí’s death in 1926.

Some of the buildings have been planned but are not yet built. Some may never be built. The design by the firm of Heneghan Peng for the Grand Egyptian Museum, the winner of a UNESCO competition of 2003, is covered with triangular Sierpiński sieves with openings at the end of galleries so that viewers can always see the Great Pyramids in the distance from inside the museum. The sieves echo the triangles on the upper halves of regular octahedra that form pyramids. The building was scheduled to open in 2013 but with the uncertainty of the political climate in the region, its future is uncertain.

The most imposing of the projects in Europe is the Bishopsgate Tower (The Pinnacle) in London, currently under construction and designed by Kohn Pedersen Fox of New York. At 63 stories it will be the second tallest building in the UK-European Union, when completed. It consists of a gigantic spiral in the form of a coiled surface. Perhaps its best feature is that when complete, at least from certain directions, it will obscure the infamous “Gherkin” (the Swiss Re Building), from the firm of Norman Foster. That gigantic green pickle rising up out of London’s financial district has been highly controversial. People love it or loathe it. Being indifferent to it is not possible. In any case, it’s not very mathematical.

The idea of living in a house that is a “Möbius strip” is perhaps a bit startling. And the description of how it works may be best left to the authors in their account of the Möbius House designed by the UN Studio in The Netherlands: it was

conceived for a young family in which to stage their diurnal, looping transition between activities on a continuous path without end. The planning, responsive to the bucolic site and daily passage of the sun, is based on a crossing, rising and falling three-dimensional path, along which the rooms and activities of the day are arranged, with the major rooms subsumed in the circular ribbon.

From the pictures it is not so easy to see how this works. It is more clear, however, in the case of the Möbius Bridge, a stunning pedestrian bridge over the Avon, designed by Hakes Associates of Bristol, UK. Though only in the planning stage, images convince readers that it would work. The authors point out that it is “prestressed by the arch on one side being set higher than intended and allowed to drop under gravity, thus increasing tension, torsion, and strength.” The reader will be relieved to know that the walkway part of the bridge does not contain the “twist” in the Möbius band, so pedestrians are not expected to walk upside down part of the way across. A whimsical sculpture of a hanging human will be attached to the underside of the pathway, to suggest that walking upside down is a possibility.

One of the more challenging structures is the Klein Bottle House in Victoria, Australia, by McBride Charles Ryan. The authors remind the reader that the figure “is a continuous, non-orientable surface that intersects itself when immersed in three dimensions… This lack of distinction between the internal and external gave expression to the designers’ intentions and organized the house into all sorts of spaces gathered around a central area, thus creating a new type of family home in which everyone has proximity to each other.” Is that really a good idea? Many photographs and drawings make it clear that this house does look something like a “Klein bottle.” But that suggests to this reader some problems, such as designing an air-conditioning system for the building, since there is no obvious distinction between the inside and the outside! The authors helpfully relate the oft-told story about “bottle” not being the right word, resulting from a mistranslation of Felix Klein’s use of Fläche (surface) as Flasche (bottle). (Unfortunately, as stated in the text the story doesn’t make much sense due to a typographical error — not something encountered here often, I’m happy to report.)

The inclusion of airports is de rigeur in books of this type and here we have the new airports at Beijing (Foster and Partners) and Abu Dhabi (Kohn Pedersen Fox), the first displaying tetrahedra and octahedra to generate patterns of equilateral triangles, the second using undulating surfaces exhibiting variations on the quatrefoil. They are huge sprawling buildings, the Abu Dhabi structure almost organic in appearance. But we’ll have to see how well they work. Airports are almost always more impressive from a distance than from the security line.

The firm of Foster and Partners is represented in adjoining chapters with, first, the great covered court at the “new” British Museum, and by the similar installation in Washington for the Smithsonian Institution: the National Portrait Gallery and the American Art Museum. These ceiling tessellations are all quite predictable and comfortable. The authors then use some diagrams effectively to make some sense of Daniel Libeskind’s wing of the Victoria and Albert Museum, the “Spiral Extension” in London.

I almost always found the mathematical explanations of the buildings understandable and helpful, but just occasionally the task of deciphering the text is challenging, as in the case of the wildly idiosyncratic design by Ashton Raggatt McDougall for the National Museum of Australia in Canberra. I was moving along just fine till I read that one of the partners in the design firm said, “This is a knot and no one can read it as a knot. Antispace is the thing.” I was just recovering from that statement when I learned that “the use of the number five in the building’s geometry is an allusion to the stigmata of Christ.” I eventually concluded that knots play a major role in the building but I don’t quite know how. The authors sum it up with: “Neither mathematics nor the digital process were starting points for the conceptual design of this building; it was only through the use of computation that these convoluted representations of meaning could adopt their knotted topology and assert their negative presence in the spatial sequence of the museum.” Accompanying pictures didn’t help.

But such flights of fantasy are the exception and overall the book is enormously satisfying. It is lavishly illustrated with photographs, diagrams, drawings and detailed plans. A glossary of mathematical terms, notes and a bibliography are provided. My only disappointment here is the lack of an index, But that is a very minor complaint about such an exciting new book for architects and mathematicians alike.

The images above, from page 20 and page 192 of the book under review, are used with the permission of the publisher.

Gerald L. Alexanderson is the Michael and Elizabeth Valeriote Professor in Science at Santa Clara University. He has served the MAA as President, Secretary, and editor of Mathematics Magazine.